Download Heat transfer across metal-dielectric interfaces during ultrafast

Liang Guo
Stephen L. Hodson
Timothy S. Fisher
Xianfan Xu1
e-mail: [email protected]
School of Mechanical Engineering
and Birck Nanotechnology Center,
Purdue University,
West Lafayette, IN 47907
Heat Transfer Across
Metal-Dielectric Interfaces
During Ultrafast-Laser Heating
Heat transfer across metal-dielectric interfaces involves transport of electrons and phonons accomplished either by coupling between phonons in metal and dielectric or by coupling between electrons in metal and phonons in dielectric. In this work, we investigate
heat transfer across metal-dielectric interfaces during ultrafast-laser heating of thin
metal films coated on dielectric substrates. By employing ultrafast-laser heating that creates strong thermal nonequilibrium between electrons and phonons in metal, it is possible
to isolate the effect of the direct electron–phonon coupling across the interface and thus
facilitate its study. Transient thermo-reflectance measurements using femtosecond laser
pulses are performed on Au–Si samples while the simulation results based on a twotemperature model are compared with the measured data. A contact resistance between
electrons in Au and phonons in Si represents the coupling strength of the direct
electron–phonon interactions at the interface. Our results reveal that this contact resistance can be sufficiently small to indicate strong direct coupling between electrons in
metal and phonons in dielectric. [DOI: 10.1115/1.4005255]
Keywords: interface thermal resistance, ultrafast laser, thermo-reflectance, two-temperature model, electron–phonon coupling
1
Introduction
Interface heat transfer is one of the major concerns in the design
of microscale and nanoscale devices. In metal, electrons, and phonons are both energy carriers while in dielectric phonons are the
main energy carrier. Therefore, for metal-dielectric composite
structures, heat can transfer across the interface by coupling
between phonons in metal and dielectric or by coupling between
electrons in metal and phonons in dielectric through electroninterface scattering. Phonon–phonon coupling has been simulated
mainly by the acoustic mismatch model and the diffuse mismatch
model [1]. As for electron–phonon coupling, there are different
viewpoints. Some studies have assumed that electron–phonon
coupling across a metal-dielectric interface is negligible and heat
transfer occurs as electron–phonon coupling within metal and
then phonon–phonon coupling across the interface [2]. Electron–
phonon coupling between metal (Cr, Ti, Al, Ni, and Pt) and SiO2
has exhibited negligible apparent thermal resistance using a
parallel-strip technique [3]. On the other hand, comparison
between simulations and transient thermal reflectance (TTR)
measurements for Au-dielectric interfaces reveals that energy
could be lost to the substrate by electron-interface scattering during ultrafast-laser heating, and this effect depends on electron
temperature and substrate thermal properties [4–6].
In this study, we employ TTR techniques to investigate interface heat transfer for thin gold films of varying thicknesses on silicon substrates. (Here, we consider silicon as a dielectric since heat
is carried by phonons in silicon.) Similar work has been reported
[5]. In our model, we consider two temperatures in metal and also
the temperature in the dielectric substrate. This allows us to investigate the effect of both the coupling between electrons in metal
and phonons in the dielectric substrate, and the coupling between
phonons in metal and phonons in the dielectric substrate, and
1
Corresponding author.
Contributed by the Heat Transfer Division of ASME for publication in the
JOURNAL OF HEAT TRANSFER. Manuscript received May 18, 2011; final manuscript
received September 30, 2011; published online February 13, 2012. Assoc. Editor:
Robert D. Tzou.
Journal of Heat Transfer
allows us to isolate the effect of the electron–phonon coupling
across the interface that can be determined from the TTR measurement. Experimentally, we employ pulse stretching to minimize the effect of nonequilibrium among the electrons. As a
result, the experimental data can be well-explained using the computational model. The thermal resistance between electrons in Au
and phonons in Si, which quantifies the direct electron–phonon
coupling strength, is calculated from the measured data. The
results reveal that in the thermal nonequilibrium state, this
electron–phonon coupling at the interface is strong enough to
dominate the overall interface heat transfer.
2
TTR Measurement
Au–Si samples of varying Au thicknesses were prepared by
thermal evaporation at a pressure of the order of 107 Torr. The
thicknesses of the gold films are 39, 46, 60, 77, and 250 nm, measured using an atomic force microscope. The pump-and-probe
technique is used in a collinear scheme to measure the thermoreflectance signal. The laser pulses are generated by a Spectra
Physics Ti:Sapphire amplified femtosecond system with a central
wavelength of 800 nm and a repetition rate of 5 kHz. The wavelength of the pump beam is then converted to 400 nm with a second harmonic crystal. The pump pulse has a pulse width (full
width at half maximum-FWHM) of 390 fs measured by the sumfrequency cross-correlation method and is focused onto the sample with a spot radius of 20.3 lm. The probe beam has a central
wavelength of 800 nm and a pulse width of 205 fs measured by
autocorrelation and is focused with a spot radius of 16.9 lm. This
pump pulse width is intentionally stretched from the original pulse
width of 50 fs to minimize the influence of thermal nonequilibrium among electrons since the electron thermalization time in
Au can be of the order of 100 fs [7]. This thermalization time is
pump wavelength and pump fluence dependent, and can be of the
order of 10 fs if higher laser fluence is used [8,9]. Our experiments
did show the importance of pulse stretching. Figure 1 shows the
TTR measurement results for the sample of thickness 77 nm with
different pump fluences before and after stretching the pulse. The
plots show the normalized relative reflectance change (DR/R)
C 2012 by ASME
Copyright V
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Cp
@Tp
@ 2 Tp
¼ kp 2 þ GðTe Tp Þ
@t
@x
@Ts
@ 2 Ts
¼ ks 2
@t
@x
Te ðt ¼ 0Þ ¼ Tp ðt ¼ 0Þ ¼ Ts ðt ¼ 0Þ ¼ T0
@Te Te Ts ke
¼
@x x¼L
Res x¼L
@Tp Tp Ts kp
¼
@x x¼L
Rps x¼L
@Ts Te Ts Tp Ts ks
¼
þ
@x x¼L
Res x¼L
Rps x¼L
Cs
Fig. 1 TTR measurement results for the Au–Si sample of Au
thickness 77 nm with different fluences. (a) Results before
pulse stretching; (b) results after pulse stretching.
with the delay time between the pump and the probe pulses to
show the contrast in cooling rates. With a shorter pulse (Fig.
1(a)), a steep initial drop is seen in the signal, which is attributed
to the behavior of nonequilibrium among electrons. Since the
TTM to be used for simulation assumes a well-defined temperature for electrons, i.e., the electrons in gold have reached thermal
equilibrium (not necessarily a uniform temperature), the model
cannot predict the fast initial drop in the signals in Fig. 1(a). As
will be shown later, the signals obtained by stretching the pulse
can be predicted well using the TTM.
3 Two-Temperature Model for Thermal Reflectance
Measurements
Ultrafast-laser heating induces thermal nonequilibrium between
electrons and phonons in metal, which can be described by the
TTM [10–13]. We note that the heterogeneous interface considered here involves three primary temperature variables (two in the
metal and one in the dielectric). The “two-temperature” model is
applied to the metal side. For investigating electron–phonon and
phonon–phonon coupling at the interface, two thermal resistances
are defined: Res (its reciprocal) indicates the coupling strength
between electrons in metal and phonons in dielectric, while Rps
indicates the coupling strength between phonons in metal and
phonons in dielectric. (Large thermal resistance corresponds to
weak coupling.) The resulting governing equations, initial, and
interface conditions are
Ce
@Te
@ 2 Te
¼ ke 2 GðTe Tp Þ þ S
@t
@x
042402-2 / Vol. 134, APRIL 2012
(1a)
(1b)
(1c)
(2)
(3a)
(3b)
(3c)
The subscripts e, p, and s denote electrons in metal, phonons in
metal, and phonons in the dielectric substrate, respectively. C is
the volumetric heat capacity, k is the thermal conductivity, G is
the electron–phonon coupling factor governing the rate of energy
transfer from electrons to phonons in metal, and L is the thickness
of the metal layer. At the front surface of the metal layer insulation boundary condition is used due to the much larger heat flux
caused by laser heating relative to the heat loss to air. At the rear
surface of the substrate, since the thickness of the substrate used is
large enough (1 lm) so that there is no temperature rise during the
time period of consideration, the insulation boundary condition is
also applied. Thermal properties of phonons in both metal and
dielectric are taken as temperature-independent due to the weak
temperature dependence. The thermal conductivity of phonons in
metal is much smaller than that of the electrons and is taken in
this work as 0.001 times the bulk thermal conductivity of gold
(311 W/(mK)). The volumetric heat capacity of the metal phonon
is taken as that of the bulk gold. Ce is taken as proportional to Te
[14] with the proportion coefficient being 70 J/(m3K2) [15], and ke
is calculated by the model and the data used in Ref. [13] which is
valid from the room temperature to the Fermi temperature
(6.39 104 K in Au, [14]). G can be obtained using the model
derived in Ref. [16]. In this work, the value of G at the room temperature is taken as 4.6 1016 W/(m3K) [17], and its dependence
on electron and phonon temperatures follows [16]. The laser heating source term S is represented by the model used in [13] as
"
2 #
0:94ð1 RÞJ
x
t
exp 2:77
S¼
L
d þ db
tp
tp ðd þ db Þ 1 exp d þ db
(4)
which assumes all the absorbed laser energy is deposited in the
metal layer. J is the fluence of the pump laser, R is the surface reflectance to the pump, tp is the pulse width (FWHM), d is the optical penetration depth, and db is the electron ballistic length
(around 100 nm in Au, [18]). Res and Rps are treated as free parameters for fitting the experimental data.
The wavelength of the probe laser in the experiment is centered
at 800 nm. For this wavelength, the incident photon energy is
below the interband transition threshold in Au, which is around
2.47 eV [18], and the Drude model can be used to relate the temperatures of electrons and phonons to the dielectric function and
then the index of refraction, which is expressed as [19]
e ¼ e1 x2p
xðx þ ixs Þ
(5)
x is the frequency of the probe laser and xp is the plasma frequency (1.37 1016 rad/s in Au evaluated using the data in Ref.
[14]). xs is the electron collisional frequency, the inverse of the
electron relaxation time. The temperature dependence of electrical
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Fig. 2 TTR measurement results on Au–Si samples of varying
Au thicknesses
resistivity indicates that xs is approximately proportional to phonon temperature at high temperature [14] and from the Fermi liquid theory, its variation with electron temperature is quadratic
(T2e ) [20]. Therefore, xs is related to Te and Tp approximately as
xs ¼ Aee Te2 þ Bep Tp
(6)
Aee is estimated from the low-temperature measurement [21] and
Bep is usually estimated from the thermal or electrical resistivity
near the room temperature [14]. In this work, Aee is taken as the literature value 1.2 107 s1K2 [6] while e1 and Bep are evaluated
by fitting the room-temperature value of the complex dielectric constant at 800 nm wavelength provided in Ref. [22], which are found
to be 9.7 and 3.6 1011 s1K1, respectively. The complex index
of refraction n0 þ in00 is the square root of the dielectric constant.
Using Eqs. (5) and (6), n0 and n00 are evaluated as 0.16 and 4.90,
respectively, which agree with the empirical values [23]. The reflectance is then calculated from n0 and n00 by the method of transfer
matrix [24], which considers multiple reflections in thin films.
4
Results and Discussion
The results of TTR measurements with a pump fluence of
147 J/m2 are plotted in Fig. 2. The fast decrease of the reflectance
indicates that energy transfer between electrons and phonons in
metal, followed by a relatively slow decrease after several ps
which indicates electrons and phonons have reached thermal equilibrium. The initial cooling rates are smaller for samples with
thicknesses less than the electron ballistic length since the electron
temperature is almost uniform across the thin film, and coupling
with phonons within the metal film and the dielectric substrate is
the only cooling mechanism. For a thicker sample of thickness
250 nm, the initial decrease is much faster due to thermal diffusion in the gold film caused by a gradient of the electron temperature in the film.
We investigate the effect of Res and Rps using the thermoreflectance signal. Two values of Rps, 1 1010 m2K/W and
1 107 m2K/W, are used, each with a parameterized range of
values for Res. Figure 3 shows the calculated results for the sample
with a 39 nm-thick gold film.
Little difference can be seen between Figs. 3(a) and 3(b) while
different cooling rates are obtained with varying Res in either plot,
indicating that the cooling rate is not sensitive to the coupling
strength between phonons in metal and dielectric. Note that an
interface resistance of 1 1010 m2K/W is lower than any
reported value, indicating a very high coupling strength between
the phonons in metal and dielectric. Conversely, the results vary
greatly with the coupling strength between electrons in metal and
phonons in dielectric at the interface. This is because the lattice
Journal of Heat Transfer
Fig. 3 Simulation results with varying Res for the Au–Si sample
of Au thickness 39 nm. (a) Rps 5 1 3 10210 m2K/W; (b) Rps
5 131027 m2K/W.
(phonon) temperature rise in metal is much smaller than the electron temperature that the interface coupling between phonons in
metal and dielectric does not influence the surface temperature,
which directly determines the measured reflectance. On the other
hand, the temperature rise of electrons is much higher, and consequently, the cooling rate is sensitive to Res. The relatively high
sensitivity of Res to that of Rps demonstrates that the former can
be isolated for the study of the coupling between electrons in
metal and phonons in dielectric.
We now use the measured TTR data to estimate Res, the thermal
resistance between electrons in metal and phonons in dielectric.
Res is adjusted by the least square method to fit the simulation
results with the measured data, and the results are shown in Fig. 4.
We note that it is impossible to fit the measured results using insulation interface condition (i.e., no coupling or extremely large
thermal resistance between electrons in metal and phonons in the
dielectric substrate), which will significantly underestimate the
cooling rate. For thin samples, we find that the value of Res is of
the order of 1010 to 109 m2K/W. This value is below the thermal resistances of representative solid–solid interfaces measured
in thermal equilibrium [25]. This indicates that the direct coupling
between electrons in metal and phonons in dielectric is strong. It
is also noted that the resistance values increases with the thickness
of the gold film, indicating a decrease in the coupling strength
between electrons in metal and the dielectric substrate. This could
be due to the lower electron temperature obtained in thicker films,
and a decrease of the coupling strength with a decrease in the
electron temperature [5]. For the sample of thickness 250 nm, Res
has little effect on the simulation result since the interface is too
far from the absorbing surface to influence the surface temperature, and therefore it is not presented here.
The agreement between the fitted results and the measured data
is generally good. The small discrepancy between the measured
and the fitted results can result from inaccuracy in computing the
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Fig. 4 Comparison between the measurement and the simulation results for
Au–Si samples of different Au thicknesses. The open circle represents the measured data and the solid line represents the simulation results. (a) 39 nm fitted by
Res 5 5 3 10210 m2K/W; (b) 46 nm fitted by Res 5 6 3 10210 m2K/W; (c) 60 nm fitted
by Res 5 1.2 3 1029 m2K/W; and (d) 77 nm fitted by Res 5 1.8 3 1029 m2K/W.
absorption or the temperature. Figure 1(b) shows the normalized
TTR measurement results on the sample of thickness 77 nm with
three laser fluences. It is seen that small variations in the shape of
the TTR signals can be caused by different laser fluences and thus
the maximum temperature reached in the film. Absorption in metal,
multiple reflections between the metal surface and the Au–Si interface, and possible deviations of the properties of thin films from
those of bulk can all contribute to uncertainties in the temperature
simulation; therefore affecting the calculated reflectance.
With the values of Res shown in Fig. 4, the calculation shows
that the highest electron temperature, which is at the surface of 39
nm–thick gold film, is about 6700 K. The highest temperature of
electrons is roughly inversely proportional to the thickness of the
films for the four thinner films. The highest temperature of electrons is much less than the Fermi temperature and thus ensures the
validity of the linear dependence of Ce on Te [14]. The highest
temperature for the lattice in metal is about 780 K, also in the 39
nm-thick gold film. This large temperature difference between
electrons and lattice indicates that the interface heat transfer is
dominated by the coupling between electrons in metal and the
phonons in the dielectric substrate. As shown in Fig. 4, the measured Res is very low, of the order of 1010 to 109 m2K/W. Even
if Rps, which is not determined in this study, is also that low (note
that 1010 to 109 m2K/W is lower than any reported values),
because of the large difference in temperatures between electrons
and the phonons in metal, the interface heat transfer rate (Eqs.
(3a)–(3c)) due to the coupling between electrons in metal and the
substrate is much larger than that due to the coupling between
phonons in metal and the substrate.
5
Conclusions
In conclusion, TTR measurements using femtosecond laser
pulses are performed on Au–Si samples and the results are
042402-4 / Vol. 134, APRIL 2012
analyzed using the TTM model. It is shown that due to the strong
nonequilibrium between electrons and phonons during ultrafastlaser heating, it is possible to isolate the effect of the direct
electron–phonon coupling across the interface, allowing investigation of its strength. Using stretched femtosecond pulses is shown
to be able to minimize the nonequilibrium effect among electrons,
and is thus more suitable for this study. The TTR measurement
data can be well-represented using the TTM model. Comparison
between the TTR data and the TTM results indicates that the
direct coupling due to electron-interface scattering dominates the
interface heat transfer during ultrafast-laser heating of thin films.
Acknowledgment
This paper is based upon work supported by the Defense
Advanced Research Projects Agency and SPAWAR Systems Center, Pacific under Contract No. N66001-09-C-2013. The authors
also thank C. Liebig, Y. Wang, and W. Wu for helpful
discussions.
Nomenclature
Aee ¼
Bep ¼
C¼
G¼
i¼
J¼
k¼
L¼
n0 ¼
n00 ¼
R¼
S¼
coefficient in Eq. (6), s1K2
coefficient in Eq. (6), s1K1
volumetric heat capacity, J/(m3K)
electron–phonon coupling factor, W/(m3K)
unit of the imaginary number
fluence of the pump, J/m2
thermal conductivity, W/(mK)
metal film thickness, m
real part of the complex index of refraction
imaginary part of the complex index of refraction
interface thermal resistance, m2K/W; reflectance
laser source term, W/m3
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T¼
t¼
tp ¼
x¼
e¼
e1 ¼
d¼
db ¼
x¼
xp ¼
xs ¼
temperature, K
time, s
pulse width of the pump (FWHM), s
spatial coordinate, m
complex dielectric constant
constant in the Drude model
radiation penetration depth, m
electron ballistic depth, m
angular frequency of the probe, rad/s
plasma frequency, rad/s
electron collisional frequency, rad/s
Subscripts
0¼
e¼
es ¼
p¼
ps ¼
s¼
initial state
electron in metal
electron in metal and phonon in dielectric
phonon in metal
phonon in metal and phonon in dielectric
phonon in dielectric
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